150 UNIVERSITY OF COLORADO STUDIES 



Designating the co-ordinates of the point of intersection N of the 



tangents of the second branch of the compound curve by a, $, from the 



figure 



NT = S= {a'— a) 2 + (£' — b)\ 



According to (21) 



2da — a 2 — b 2 

 a' — a 



£' — 6 = 



2d — 2a. — 2bk ' 



— k(2aa — a 2 —b 2 ) 



2(L — 2a, — 2bk 



here k= — cot <f>. Substituting these values for a' — a and ft' — b 

 in the expression for S, and putting a = R l = O x C 1} ft = o, and redu- 

 cing, we get 



7V + 7Y - 2 T z (R z sin<fr - T z cos0) 

 ^ 2 r 2 -2(i? I sin<|.-r i cos <f>) ' [3 ° } 



According to formulas (27), the co-ordinates of the center of a part 

 of a compound curve, or reversed curve below tangent V iy Fig. 3, are 



a" = a — pk, 

 0" = &-/*, (31) 



and the radius i? 2 = pv 1 + & 2 . 



The tangency of the branches of a compound or reversed curve 

 implies the condition : 



(a - a") 2 + (0 - /3") 2 = (£, ± £,)», (32) 



in which the positive sign on the right side gives a reversed curve,*and]the 



negative sign a compound curve. 



Substituting the values for a", /3", R 2 given in (31) in"(32),we"obtain 



for /* the expression : 



2R J a — a 2 — b 2 



P = / \ ' - 



2 R.ik + V 1 + k 2 ) — 2 (ak + b) 



and replacing a, b, k by T 2 sin <£, 7\ + T 2 cos <£, — cot <j>; 



T T 2 + T 2 2 - 2 T 2 (R, sin <j> - T x cos» m 



H = : — 7 —jk t 



■2 



sin<£ 



\ ±R l + T l sm<f> + R, cos<£ [ 



